I would tackle the problem that way. Calculate the principal strains and add them together.

EDIT: After looking at the problem, it is in a state of zero shear. Therefore [tex]\sigma_x[/tex], [tex]\sigma_y[/tex] and [tex]\sigma_z[/tex] are going to be the principal stresses. So, if you calculate the strains from the principal stresses you will be calculating the principal strains. Things should be pretty straight forward. I ended up calculating [tex]6.5 x 10^{-4}[/tex]

Do you know the 3 dimensional stress-strain relations that involve E and [tex]\nu[/tex]?